# GMAT Math : Circles

## Example Questions

### Example Question #1 : Dsq: Calculating The Length Of A Radius

Give the radius of a circle on the coordinate plane.

Statement 1: A square whose vertices include  and  can be inscribed inside the circle.

Statement 2: A right triangle whose vertices include  and  can be inscribed inside the circle.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. The length of a segment with the given endpoints can be calculated using the distance formula. However, it is not clear whether the points are opposite vertices, in which case the segment is a diagonal of the square, or the points are consecutive vertices, in which case the segment is a side of the square, making the diagonal of the square  times this length. The length of the diagonal of the inscribed square cannot be determined for certain; since the diameter of the circle is equal to the length of the diagonal, the diameter cannot be determined, and since the radius is half this, the radius cannot be determined.

Assume Statement 2 alone. The length of a segment with the endpoints can be calculated using the distance formula. However, it is not clear whether the segment is a hypotenuse of the triangle or not; the diameter of a circle is equal to the length of the hypotenuse of an inscribed right triangle, so knowing this is necessary.

Assume both statements to be true. The two statements together give four points of the circle; since three points uniquely define a circle, the circle can be located; subsequently, the radius can be found.

### Example Question #2 : Dsq: Calculating The Length Of A Radius

Rectangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The diameter of a circle that circumscribes a rectangle is equal to the length of a diagonal of the rectangle; the radius is equal to half this.

The two statements together, however, do not yield this. The opposite sides of a rectangle are congruent, so the two statements are actually equivalent; each gives the same dimension of the rectangle. This is insufficient to determine the length of the diagonal.

### Example Question #3 : Dsq: Calculating The Length Of A Radius

An equilateral triangle  is inscribed inside a circle;  is the midpoint of . What is the radius of the circle?

Statement 1:  has area .

Statement 2: .

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

First, locate the other midpoints of the sides of the triangle and construct the segments from each vertex to the opposite midpoint.

Since  is equilateral, , and  are all altitudes that insersect at the center of the circumscribed circle, , so that  is the radius of the circumscribed circle.

Assume Statement 1 alone. The length of one side of an equilateral triangle can be calculated using the formula

, or, equivalently,

Once  is calculated, then, since   is also a perpendicular bisector of  and a bisector of , making  a 30-60-90 triangle,  can be calculated to be one half of  can be multiplied by  to yield , and, since the three altitudes of an equilateral triangle divide one another into segments whose lengths have ratio 2:1,  can be multiplied by  to obtain radius .

Statement 2 gives us  explicitly, so we can take two thirds of this to get the radius

.

### Example Question #4 : Dsq: Calculating The Length Of A Radius

Between Circle 1 and Circle 2, which has the greater radius?

Statement 1: A  arc of Circle 1 has length equal to one fourth the circumference of Circle 2.

Statement 2: A  sector of Circle 2 has area equal to four ninths that of Circle 1.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A  arc of Circle 1 has  the circumference of Circle 1. Since this is also  the circumference of Circle 2, then, if we let  be the circumferences,

,

and

.

This gives Circle 2 the greater circumference and, subsequently, the greater radius.

Assume Statement 2 alone. A  sector of Circle 2 has area  of Circle 2. Since its area is also equal to  that of Circle 1, then, if  are the areas of Circle 1 and Circle 2, respectively, then

,

and

This gives Circle 2 the greater area and, subsequently, the greater radius.

### Example Question #5 : Dsq: Calculating The Length Of A Radius

Give the radius of a circle on the coordinate plane.

Statement 1: A right triangle with a hypotenuse with endpoints  and  can be inscribed in the circle.

Statement 2: A right triangle with a leg with endpoints  and  can be inscribed in the circle.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

If a right triangle can be inscribed inside a given circle, then its hypotenuse has a length equal to the diameter of the circle, and the radius of the circle can be calculated as half this. Statement 1 gives sufficient information to find this, since the length of the hypotenuse is the distance between its endpoints  and , which is ; the diameter of the circle is 20, and the radius is half this, or 10. From Statement 2, we can only find the length of one leg of an inscribed right triangle, so the length of the hypotenuse is still open to question.

### Example Question #441 : Geometry

Parallelogram  is inscribed inside a circle. What is the radius of the circle?

Statement 1: Each side of Parallelogram  has length 20.

Statement 2: .

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.

Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length  times that of a side, or ; half this, or , is the radius of the circle.

Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.

### Example Question #12 : Circles

A circle is inscribed inside an equilateral triangle . , and  are tangent to the circle at the points , and , respectively. What is the radius of the circle?

Statement 1: The length of arc  is .

Statement 2: The degree measure of arc  is .

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The figure referenced is below:

By symmetry,  is one third of a circle. Therefore, its length is one third of the circumference, so, if Statement 1 alone is assumed, the circumference can be determined to be ; this can be divided by  to yield radius .

Statement 2 yields no helpful information; from the body of the problem,  can already be deduced to be two thirds of a circle, or, equivalently, an arc of measure

.

### Example Question #11 : Circles

Square  is inscribed inside a circle. What is the radius of the circle?

Statement 1: Square  has area 100.

Statement 2: .

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 2 alone, , a diagonal of the square, measures . The diameter of the circle is equal to the length of a diagonal of an inscribed square, so the radius of the circle is equal to half this, or .

From Statement 1 alone, since the area of the square is 100, its sidelength is the square root of this, or 10. By the 45-45-90 Theorem, a diagonal of the square measures  times this, or , which makes Statement 2 a consequence of Statement 1. Therefore, it follows again that the circle has radius .

### Example Question #11 : Circles

Right triangle  is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.

Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.

Assume both statements are true. Since  in right triangle , then either  and , or vice versa. In either event, , being opposite the  angle, is the short leg of a 30-60-90 triangle, and, by the 30-60-90 Theorem, the hypotenuse is twice its length. This is twice 18, or 36. This is the diameter of the circle, and the radius is half this, or 18.

### Example Question #15 : Circles

Give the radius of a circle on the coordinate plane.

Statement 1: The circle has its center at .

Statement 2: The circle has its -intercepts at  and .

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.